Anti-deSitter universe dynamics in LQC

A model for a flat isotropic universe with a negative cosmological constant $\Lambda$ and a massless scalar field as sole matter content is studied within the framework of Loop Quantum Cosmology. By application of the methods introduced for the model with $\Lambda=0$, the physical Hilbert space and the set of Dirac observables are constructed. As in that case, the scalar field plays here the role of an emergent time. The properties of the system are found to be similar to those of the $k=1$ FRW model: for small energy densities, the quantum dynamics reproduces the classical one, whereas, due to modifications at near-Planckian densities, the big bang and big crunch singularities are replaced by a quantum bounce connecting deterministically the large semiclassical epochs. Thus in Loop Quantum Cosmology the evolution is qualitatively cyclic.Comment: Revtex4, 29 pages, 20 figures, typos correcte

Quantum Nature of the Big Bang: An Analytical and Numerical Investigation

Analytical and numerical methods are developed to analyze the quantum nature of the big bang in the setting of loop quantum cosmology. They enable one to explore the effects of quantum geometry both on the gravitational and matter sectors and significantly extend the known results on the resolution of the big bang singularity. Specifically, the following results are established for the homogeneous isotropic model with a massless scalar field: i) the scalar field is shown to serve as an internal clock, thereby providing a detailed realization of the `emergent time' idea; ii) the physical Hilbert space, Dirac observables and semi-classical states are constructed rigorously; iii) the Hamiltonian constraint is solved numerically to show that the big bang is replaced by a big bounce. Thanks to the non-perturbative, background independent methods, unlike in other approaches the quantum evolution is deterministic across the deep Planck regime. Our constructions also provide a conceptual framework and technical tools which can be used in more general models. In this sense, they provide foundations for analyzing physical issues associated with the Planck regime of loop quantum cosmology as a whole.Comment: Revised version to appear in Physical Review D. References added and typos correcte

Loop quantum cosmology of k=1 FRW models

The closed, k=1, FRW cosmology coupled to a massless scalar field is investigated in the framework of loop quantum cosmology using analytical and numerical methods. As in the k=0 case, the scalar field can be again used as emergent time to construct the physical Hilbert space and introduce Dirac observables. The resulting framework is then used to address a major challenge of quantum cosmology: resolving the big-bang singularity while retaining agreement with general relativity at large scales. It is shown that the framework fulfills this task. In particular, for states which are semi-classical at some late time, the big-bang is replaced by a quantum bounce and a recollapse occurs at the value of the scale factor predicted by classical general relativity. Thus, the `difficulties' pointed out by Green and Unruh in the k=1 case do not arise in a more systematic treatment. As in k=0 models, quantum dynamics is deterministic across the deep Planck regime. However, because it also retains the classical recollapse, in contrast to the k=0 case one is now led to a cyclic model. Finally, we clarify some issues raised by Laguna's recent work addressed to computational physicists

Quasi-local rotating black holes in higher dimension: geometry

With a help of a generalized Raychaudhuri equation non-expanding null surfaces are studied in arbitrarily dimensional case. The definition and basic properties of non-expanding and isolated horizons known in the literature in the 4 and 3 dimensional cases are generalized. A local description of horizon's geometry is provided. The Zeroth Law of black hole thermodynamics is derived. The constraints have a similar structure to that of the 4 dimensional spacetime case. The geometry of a vacuum isolated horizon is determined by the induced metric and the rotation 1-form potential, local generalizations of the area and the angular momentum typically used in the stationary black hole solutions case.Comment: 32 pages, RevTex

Multipole Moments of Isolated Horizons

To every axi-symmetric isolated horizon we associate two sets of numbers, $M_n$ and $J_n$ with $n = 0, 1, 2, ...$, representing its mass and angular momentum multipoles. They provide a diffeomorphism invariant characterization of the horizon geometry. Physically, they can be thought of as the `source multipoles' of black holes in equilibrium. These structures have a variety of potential applications ranging from equations of motion of black holes and numerical relativity to quantum gravity.Comment: 25 pages, 1 figure. Minor typos corrected, reference adde

Black hole boundaries

Classical black holes and event horizons are highly non-local objects, defined in relation to the causal past of future null infinity. Alternative, quasilocal characterizations of black holes are often used in mathematical, quantum, and numerical relativity. These include apparent, killing, trapping, isolated, dynamical, and slowly evolving horizons. All of these are closely associated with two-surfaces of zero outward null expansion. This paper reviews the traditional definition of black holes and provides an overview of some of the more recent work on alternative horizons.Comment: 27 pages, 8 figures, invited Einstein Centennial Review Article for CJP, final version to appear in journal - glossary of terms added, typos correcte

Trapped and marginally trapped surfaces in Weyl-distorted Schwarzschild solutions

To better understand the allowed range of black hole geometries, we study Weyl-distorted Schwarzschild solutions. They always contain trapped surfaces, a singularity and an isolated horizon and so should be understood to be (geometric) black holes. However we show that for large distortions the isolated horizon is neither a future outer trapping horizon (FOTH) nor even a marginally trapped surface: slices of the horizon cannot be infinitesimally deformed into (outer) trapped surfaces. We consider the implications of this result for popular quasilocal definitions of black holes.Comment: The results are unchanged but this version supersedes that published in CQG. The major change is a rewriting of Section 3.1 to improve clarity and correct an error in the general expression for V(r,\theta). Several minor errors are also fixed - most significantly an incorrect statement made in the introduction about the extent of the outer prison in Vaidya. 17 pages, 2 figure

Quantum geometry and the Schwarzschild singularity

In homogeneous cosmologies, quantum geometry effects lead to a resolution of the classical singularity without having to invoke special boundary conditions at the singularity or introduce ad-hoc elements such as unphysical matter. The same effects are shown to lead to a resolution of the Schwarzschild singularity. The resulting quantum extension of space-time is likely to have significant implications to the black hole evaporation process. Similarities and differences with the situation in quantum geometrodynamics are pointed out.Comment: 31 pages, 1 figur

Phase-space and Black Hole Entropy of Higher Genus Horizons in Loop Quantum Gravity

In the context of loop quantum gravity, we construct the phase-space of isolated horizons with genus greater than 0. Within the loop quantum gravity framework, these horizons are described by genus g surfaces with N punctures and the dimension of the corresponding phase-space is calculated including the genus cycles as degrees of freedom. From this, the black hole entropy can be calculated by counting the microstates which correspond to a black hole of fixed area. We find that the leading term agrees with the A/4 law and that the sub-leading contribution is modified by the genus cycles.Comment: 22 pages, 9 figures. References updated. Minor changes to match version to appear in Class. Quant. Gra

Supersymmetric isolated horizons

We construct a covariant phase space for rotating weakly isolated horizons in Einstein-Maxwell-Chern-Simons theory in all (odd) $D\geq5$ dimensions. In particular, we show that horizons on the corresponding phase space satisfy the zeroth and first laws of black-hole mechanics. We show that the existence of a Killing spinor on an isolated horizon in four dimensions (when the Chern-Simons term is dropped) and in five dimensions requires that the induced (normal) connection on the horizon has to vanish, and this in turn implies that the surface gravity and rotation one-form are zero. This means that the gravitational component of the horizon angular momentum is zero, while the electromagnetic component (which is attributed to the bulk radiation field) is unconstrained. It follows that an isolated horizon is supersymmetric only if it is extremal and nonrotating. A remarkable property of these horizons is that the Killing spinor only has to exist on the horizon itself. It does not have to exist off the horizon. In addition, we find that the limit when the surface gravity of the horizon goes to zero provides a topological constraint. Specifically, the integral of the scalar curvature of the cross sections of the horizon has to be positive when the dominant energy condition is satisfied and the cosmological constant $\Lambda$ is zero or positive, and in particular rules out the torus topology for supersymmetric isolated horizons (unless $\Lambda<0$) if and only if the stress-energy tensor $T_{ab}$ is of the form such that $T_{ab}\ell^{a}n^{b}=0$ for any two null vectors $\ell$ and $n$ with normalization $\ell_{a}n^{a}=-1$ on the horizon.Comment: 26 pages, 1 figure; v2: typos corrected, topology arguments corrected, discussion of black rings and dipole charge added, references added, version to appear in Classical and Quantum Gravit